Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1909 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12988 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #386 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1869 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
466 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,17,12,15,29,32,13,16,30,34,10,27,19,36)(8,18,22,28,23,33,37,9,21,25,35,24,20,14,31,26);(6,8,19,31,10,20,30,35,13,21,29,37,12,23,11,22)(7,14,32,28,27,25,17,26,16,33,36,24,15,18,34,9);(6,9)(7,8)(10,28)(11,18)(12,24)(13,26)(14,19)(15,23)(16,21)(17,35)(20,27)(22,34)(25,30)(29,33)(31,32)(36,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5976);